Extremal clustering and cluster counting for spatial random fields

Abstract: We consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand. Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set $C$, potentially with different scalars in different directions, we use two cluster definitions to define associated cluster counting point processes on the rescaled index set $C$; one cluster definition divides the index set into more and more boxes and counts a box as a cluster if it contains an extremal observation. The other cluster definition that is more intuitive considers extremal points to be in the same cluster, if they are close in distance. We show that both cluster point processes converge to a Poisson point process on $C$. Additionally, we find a limit of the mean cluster size. Finally, we pay special attention to the case without clusters.